C1 chapter 2 - quadratic functions

In this chapter you will learn how to plot and sketch quadratics, factorise quadratics, complete the square and use the quadratic equation and its discriminant.

C1 section 2.1 plotting quadratic graphs

By the end of this section you should be able to:

• plot any quadratic equation by working out values of coordinates and plotting them on axes.

Before you start you need to know:

• how to evaluate simple functions including those involving squaring and those using negative numbers

Examples:

1. plot the graph of the function:

$\displaystyle y = x^2 - 5x + 6$
solution: first draw up a table and work out the values, step-by-step:

then plot a graph using your values for x and y as the coordinates of the points:

C1 Exercise 2A

• C1 Exercise 2A worked solutions: please note the advice about drawing smooth curves on the last couple of pages.

• Questions similar to this exercise:

Explore graphing quadratics with this widget:






... or use the Desmos Graphing Calculator example that has some neat sliders to explore the effect of changing a, b or c in the general quadratic:

C1 section 2.2 - solving quadratics by factorisation

By the end of this section you should be able to:

• factorise suitable quadratic expressions and hence solve quadratic equations of the three types below:

1. simple quadratics of the form ax2 + bx + c = 0 where a = 1, b = (p + q) and c = pq for some integers p, q which you need to find.
   Example:

$\displaystyle x^2 + 5x + 6 = 0$
$\Rightarrow (x + 3)(x+2) = 0$
$\Rightarrow x = -3 or x = -2$

Before you start you need to know:

• how to evaluate simple functions including those involving squaring and those using negative numbers

Discussion and practice: a Tarsia jigsaw on solving quadratics by factorising:

Exercise 2B

worked solutions:

Section 2.3 Completing the Square

What's the point?
Exactly - what is the point or vertex of a parabola?
One method of finding the minimum (or maximum) of a quadratic curve is to use the completed square form or 'vertex form' of the quadratic function.
Explore the vertex form of a quadratic with this Desmos graph:

Here's the quadratic we're used to:
$\displaystyle y = ax^2 + bx + c$
But in vertex form it has been replaced by
$\displaystyle y = a(x - h)^2 + k$
So the obvious question is how do you get from one to the the other?

Exercise 2C

worked solutions:

Section 2.4

You need to learn:

Exercise 2D

worked solutions:

Section 2.5

You need to learn:

Exercise 2E

worked solutions:

Section 2.6

You need to learn:

Exercise 2F

worked solutions:
By now you should be able to recognise a variety of quadratic graphs, and although we do more work on linear graphs in the next chapter, you can probably already identify which are which from standard equations. Try the 'big function graph puzzle' to see how your graph recognition skills are coming on:

C1 chapter 2 revision